Involutory Hopf group-coalgebras and invariants of flat bundles over 4-manifolds

Nicolas Bridges; Shawn Cui

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arXiv · 2503.08861

Involutory Hopf group-coalgebras and invariants of flat bundles over 4-manifolds

Nicolas Bridges·Shawn Cui

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Abstract

We give invariants of flat bundles over 4-manifolds generalizing a result by Chaidez, Cotler, and Cui (Alg. \& Geo. Topology '22). We utilize a structure called a Hopf $G$-triplet for $G$ a group, which generalizes the notion of a Hopf triplet by Chaidez, Cotler, and Cui. In our construction, we present flat bundles over 4-manifolds using colored trisection diagrams: a direct analogue of colored Heegaard diagrams as described by Virelizier. Our main result is that involutory Hopf $G$-triplets of finite type yield well-defined invariants of $G$-colored trisection diagrams, and that if the monodromy of a flat bundle has image in $G$ we obtain invariants of flat bundles. We also show that a special Hopf $G$-triplet yields the invariant from Hopf $G$-algebras described by Mochida, thus generalizing the construction.

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BibTeX RIS

Nicolas Bridges, Shawn Cui. 2025-03-11. Involutory Hopf group-coalgebras and invariants of flat bundles over 4-manifolds. https://arxiv.org/abs/2503.08861

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